|
%I #36 Nov 09 2018 21:22:12
%S 4,6,9,10,14,15,21,22,25,26,33,34,38,39,46,49,51,57,58,62,65,69,74,82,
%T 85,86,87,91,93,94,106,111,118,121,122,123,129,133,134,141,142,145,
%U 146,158,159,166,169,177,178,183,185,194,201,202,205,206,213,214,217,218,219,226,237,249,254,259,262,265,267,274,278,289,291,298,301,302,303,305,309,314,321,326,327,334,339,341,346,358,361,362,365,381,382,386,393,394,398,411,417,422,427,445,446,447,451,453,454,458,466,469,471,478,481,482,485,489,501,502,505,511,514,519,526,529,537,538,542,543,545,553,554,561
%N Range of A000790 (primary pretenders).
%C All terms except for the last term, 561, are semiprimes (A001358). Semiprimes up to 559 that are not here: 35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 473, 493, 497, 515, 517, 527, 533, 535, 551, 559. - _Zak Seidov_, Jan 08 2015
%C The LCM of all terms is 23# * 277# (where # denotes the primorial function A034386), the period of A000790, and therefore also of the related sequence b(n) = gcd(A000790(n), n). - _M. F. Hasler_, Feb 16 2018
%C Range of A295997. - _Thomas Ordowski_, Feb 27 2018
%C These numbers k < 561 are semiprimes k = pq such that p-1 | q-1, where primes p <= q. Equivalent condition is p-1 | k-1. - _Thomas Ordowski_, Aug 18 2018
%C This shows that all even semiprimes < 561 are in this sequence. The odd semiprimes not in this sequence are the semiprimes (equivalently: all terms but 275, 455, 475, 539) less than 561 in A267999 (which equals A121707 up to 695). - _M. F. Hasler_, Nov 09 2018
%H J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, <a href="http://neilsloane.com/doc/primary.html">The Primary Pretenders</a>, Acta Arith. 78 (1997), 307-313.
%H J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0207180">The Primary Pretenders</a>, arXiv:math/0207180 [math.NT], 2002.
%t pp[n_] := For[c = 4, True, c = If[PrimeQ[c+1], c+2, c+1], If[PowerMod[n, c, c] == Mod[n, c], Return[c]]];seq[n_] := seq[n] = Table[pp[k], {k, 0, 2^n}] // Union; seq[10]; seq[n = 11]; While[ Print["n = ", n, " more terms: ", Complement[seq[n], seq[n-1]]]; seq[n] != seq[n-1], n++]; A108574 = seq[n] (* _Jean-François Alcover_, Oct 18 2013 *)
%o (PARI) my(A=List(561)); forprime(q=2,561\2, forprime(p=2,min(q,561\q), (q-1)%(p-1)|| listput(A, p*q))); A108574=Set(A) \\ _M. F. Hasler_, Nov 09 2018
%Y Cf. A000790, A001358, A295997.
%K fini,full,nonn
%O 1,1
%A _David W. Wilson_, Jun 10 2005
|
